Page 44 - Mechanical Engineers Reference Book

Mechanics of fluids 1/33

1.5.8.3 Shock waves For accurate velocity measurement with little disturbance to
the flow hot wire anemometers may be used. The resistance of
Under normal design conditions the flow in the nozzle down- an electrically heated wire is related to the temperature of the
stream of the throat will be supersonic. The velocity of the gas wire, which in turn is related to the velocity of the fluid flow
at exit will depend on the external pressurepb (back pressure). past the wire. The wire resistance measured on a bridge may
If the back pressure is greater than the theoretical exit be calibrated against a known velocity, to give either direct
pressurle pe@b > pe) then shock waves will be set up in the readout or (more usually) a calibration curve. The fine wire of
nozzle. These are discontinuities similar to standing waves in the anemometer is suceptible to fluid contamination.
open-channel flow. The shock waves set up in such a way are Other velometers and anemometers (depending on the
normal shock waves, normal to the direction of flow. relationship between the speed of rotation of a set of blades
If pb < pe then the expansion will continue outside the and the velocity (or speed) of the gas flow) may be used in
nozzle i(over-expansion). very large cross-sectional ducts or to measure wind speed in
If conditions upstream of a normal shock wave are denoted the open air. They may depend on the rotation to generate a
by suffix 1 and downstream by suffix 2, then it can be shown small electrical current, which can be calibrated as a speed, or
that the product of the up- and downstream velocities is equal the number of revolutions may be inserted into an empirical
to the square of the sonic velocity at M = 1:
formula. A typical example is the three-vaned meteorological
anemometer.
and since M, > 1, then M2 < 1. Also,
I 1.5.9 Ideal fluid flow
1. M2= (1.1 15) The concept of using idealized conditions to establish the
shape of the mathematical models of real situations is common
(1.116) in engineering science studies. These models may then be
modified to accommodate observed relationships, for applica-
tion to real situations.
(1.117) Ideal fluid (or potential) flow is such a concept. It may be
used to set up flow patterns in the region of a flow stream
These are known as the Rankine-Hugoneot relationships. outside the boundary layers described in Section 1.5.6. The
Values for air may be obtained by putting y = 1.4 or by the combination of ideal flow and the boundary layer effects may
use of published tables (Houghton and Brock, 1961). be used to predict the performance of a real situation, so long
The strength of a shock wave may be defined as the ratio of as the limitations of both are recognized. The fluid is assumed
the pressure change across the wave to the upstream pressure, to be inviscid and the flow steady, continuous and irrotational,
or in terms of the upstream Mach number: as defined in Section 1.5.3.1. This means that there are no
cavities or discontinuities in the flow stream, and that the fluid
particles do not rotate about their own axes, even though the
(1.118) flow may be circular.
strength PI The continuity equation (1.23) applies, and may be mo-
= 1.167(M12 - I), for air. dified to
Oblique shock waves; at an angle p to the upstream flow d””+!!Y=0
direction, are produced when a supersonic gas flow is turned ax ay ( 1.1 19)
through an angle 6 by an obstruction such as an aircraft’s nose,
wing or tail, the inside walls of a duct, etc. The relationships if required for two-dimensional flow. For irrotation in two
between the up- and downstream Mach numbers and the dimensions equation (1.28) becomes
angles 13 and 0 are published in tables and charts (Houghton d”* _-
and Brock, 1961). = 0 (1.120)
In some cases both the up- and downstreams will be ay ax
supersonic and subsequent shock waves produced, for
example, at the leading and trailing edges of a wing. The I S.9.1 The stream function
effects of such shock waves produced by aircraft in flight, say, From the definitions of the stream line and stream function in
may be noted at ground level as sonic booms. Section 1.5.3.l(e) the equation to a stream line may be shown
to be
1.5.8.4 Gas flow measurement vxdy - v,& = 0 (1.121)
Gas flow rates through ducts will normally be measured using and since the stream function I/I is an equation which describes
devices and techniques similar to those used for in- a family of stream lines, then, for example.
compressible fluids, namely orifice plates, venturi meters and *=2x-y
nozzles. However, for gases the flow rate is usually quoted as a
mass flow rate. Equations (1.110) and (1.112) may be used represents a family of parallel straight lines with a variable
with orifices and venturis as well as nozzles. Relevant values of intercept (I.
C, for leach device wiil be found in BS 1042, in addition to For unit thickness in the z direction the volumetric flow rate
operational advice. V between two stream lines 1 and 2 is
When Pitot-static tubes are used to measure gas flow
velocities equation (1.58) may be acceptable for low flows with (1.122)
low pressure differences. At high velocities the compressibility
must be taken into account and equation (1.110) used with
stream conditions at X replacing those at the throat t. (I. 123)